Markov Entropy Centrality: Chemical, Biological, Crime, and Legislative Networks

In this chapter, we propose the study of multiple systems using node centrality or connectedness information measures derived from a Graph or Complex Network. The information is quantified in terms of the Entropy centrality kC θ(j) of the jth parts or states (nodes) of a Markov Chain associated with the system, represented by a network graph. The procedure is standard for all systems despite the complexity of the system. First, we define the phenomena to study, ranging from molecular systems composed by single molecules (drug activity, drug toxicity), multiple molecules (networks of chemical reactions), and macromolecules (DNA–drug interaction, protein function), to ecological systems (bacterial co-aggregation), or social systems (criminal causation, legislative productivity). Second, we collect several cases from literature (drugs, chemical reactions, proteins, bacterial species, or criminal cases). Next, we classify the cases in at least two different groups (active/nonactive drugs, enantioselective/non-enantioselective reactions, functional/nonfunctional proteins, co-aggregating/non-co-aggregating bacteria, or crime/noncrime cause, efficient/nonefficient law). After that, we represent the interconnectivity of the discrete parts of the system (atoms, amino acids, reactants, bacteria species, or people) as a graph or network. The Markov Chain theory is used to calculate the entropy of the system for nodes placed at different distances. Finally, we aim to both derive and validate a classification model using the entropy values as input variables and the classification of cases as the output variables. The model is used to predict the probability with which a case presents the studied property. The present work proposes the entropy of a Markov Chain associated with a network or graph to be used as a universal quantity in pattern recognition regardless the chemical, biological, social, or other nature of the systems under study.